Muenzaniso wemubvunzo wekukurukurirana pamusoro pekushandiswa kwezvinomera
Rinobva pane rimwe ipfungwa huru yekuverenga ine mashandisirwo akawanda muhupenyu hwezuva nezuva nedzimwe nzvimbo dzesainzi, dzakadai sefizikisi, economics, biology, uye engineering. Muchinyorwa chino, tichakurukura matambudziko akati wandei uye tichakurukura mashandisirwo emaderivatives, kunyanya mukugadzirisa nekugadzirisa mashandiro.
Nhanganyaya kuZvikumbiro Zvakatorwa
Chinobva pabasa chinopa ruzivo nezve mwero wekuchinja kwebasa iroro maererano nekuchinja kwaro kwakazvimirira. Muenzaniso uri nyore ispeed, inova iyo inobva panzvimbo maererano nenguva. Zvakanyanya, ma derivatives anogona kushandiswa kuwana maximum uye minimum values ebasa, kuona nguva iyo basa riri kuwedzera kana kuderera, uye kupa ruzivo nezvehunhu uye maitiro emifananidzo ebasa.
Muenzaniso Mubvunzo 1: Kuwana Hunhu Hwakanyanya uye Hwakaderera
Mubvunzo:
Sarudza mapoinzi epamusoro uye epasi ebasa \( f(x) = x^3 – 3x^2 + 4 \).
Kukurukurirana:
1. Kutsvaga derivative yekutanga:
Kuti tiwane mapoinzi akakosha, tinofanira kuwana derivative yekutanga yebasa toienzanisa ne zero.
\[
f'(x) = 3x^2 – 6x
\]
\[
3x^2 – 6x = 0
\]
2. Gadzirisa equation:
Tinotarisa equation yacho:
\[
3x(x – 2) = 0
\]
Saka, tinowana mapoinzi akakosha pa \( x = 0 \) uye \( x = 2 \).
3. Ongorora chinobva chechipiri:
Kuti tione kana pfungwa dzinokosha dziri maxima kana kuti minima, tinofanira kuwana derivative yechipiri yebasa:
\[
f”(x) = 6x – 6
\]
Kuongorora panguva dzakakosha:
\[
f”(0) = 6(0) – 6 = -6 \, (\text{negative, saka\ } x = 0 \text{\ is a local maximum})
\]
\[
f”(2) = 6(2) – 6 = 6 \, (\text{positive, so\ } x = 2 \text{\ idiki remunharaunda})
\]
4. Verenga huwandu hwepamusoro uye hwepasi:
Tsiva mapoinzi akakosha mubasa rekutanga:
\[
f(0) = 0^3 – 3 \cdot 0^2 + 4 = 4 \, (\text{maximum})
\]
\[
f(2) = 2^3 – 3 \cdot 2^2 + 4 = 8 – 12 + 4 = 0 \, (\text{minimum})
\]
Saka, basa \( f(x) = x^3 – 3x^2 + 4 \) rine maximum yenzvimbo pa \( (0, 4) \) uye maximum yenzvimbo pa \( (2, 0) \).
Muenzaniso Mubvunzo 2: Kugadzirisa Zvipingamupinyi
Mubvunzo:
Murimi anoda kuvaka nzvimbo yakavharirwa ine mativi mana yakakomberedza rwizi. Kana wapiwa fenzi yakareba mamita zana, sarudza saizi yenzvimbo yacho kuti ukwanise kuwedzera nzvimbo yayo.
Kukurukurirana:
1. Gadzira equation:
Ngatitii kureba kwenzvimbo yakatarisana nerwizi kuri \( x \) mamita uye upamhi huri \( y \) mamita. Sezvo rumwe rutivi rwakaganhura rwizi, fenzi inodiwa ndeyemativi matatu.
\[
2y + x = 100
\]
2. Tsvaga nzvimbo yakanyanya kunaka:
Nzvimbo yekeji \( A \) ndeiyi:
\[
A = x \cdot y
\]
Kubva pakuenzanisa kwefenzi, tinogona kutaura \(\\y \) maererano ne \(x \):
\[
y = \frac{100 – x}{2}
\]
Saka, equation yenzvimbo inova:
\[
A(x) = x \cdot \frac{100 – x}{2} = 50x – \frac{x^2}{2}
\]
3. Kutsvaga derivative yekutanga:
Kuti tiwane kukosha kwakanyanya, tinowana derivative yekutanga ye \( A(x) \):
\[
A'(x) = 50 – x
\]
Kuenzana kusvika pazero:
\[
50 – x = 0 \zvinoreva x = 50
\]
4. Verenga kukosha kwe \( y \):
Isa \( x = 50 \) muequation:
\[
y = \frac{100 – 50}{2} = 25
\]
Saka, zviyero zvekeji zvinopa nzvimbo yepamusoro ndeyemamita makumi mashanu pakureba uye mamita makumi maviri nemashanu pahupamhi.
Muenzaniso Mubvunzo 3: Kuziva Kumhanya Kwakanyanya
Mubvunzo:
Chidimbu chinofamba nemutsara wakatwasuka chine nzvimbo inoratidzwa sebasa renguva \( s(t) = t^3 – 6t^2 + 9t + 1 \). Sarudza kumhanya kwakanyanya kwechidimbu.
Kukurukurirana:
1. Sarudza kumhanya (nzvimbo inobva pane chimwe chinhu):
Kumhanya kwechinhu chimwe ndicho chinhu chekutanga chinobva panzvimbo maererano nenguva:
\[
v(t) = \frac{ds}{dt} = 3t^2 – 12t + 9
\]
2. Sarudza chinobva chechipiri:
Kuti tiwane mapoinzi akanyanya, tinowana derivative yechipiri:
\[
a(t) = \frac{dv}{dt} = 6t – 12
\]
3. Kuwana pfungwa inokosha:
Kuenzanisa derivative yekutanga yevelocity ne zero:
\[
3t^2 – 12t + 9 = 0
\]
Govanisa ne3:
\[
t^2 – 4t + 3 = 0
\]
Kuenzanisa:
\[
(t – 3)(t – 1) = 0
\]
Saka, pfungwa dzakakosha ndi \( t = 1 \) uye \( t = 3 \).
4. Ongorora kukurumidza kuti uwane huwandu hwepamusoro:
\[
a(1) = 6(1) – 12 = -6 \zvinoreva t = 1 \text{\ is a local maximum}
\]
\[
a(3) = 6(3) – 12 = 6 \zvinoreva t = 3 \text{\ ipfupiso yenzvimbo}
\]
5. Kuverenga kumhanya kwakanyanya:
Isa \(t = 1 \) mu equation yekukurumidza:
\[
v(1) = 3(1)^2 – 12(1) + 9 = 3 – 12 + 9 = 0 \, (\mashoko{haafadzi})
\]
Tarisa mimwe miganhu yakakodzera kana nzvimbo dzepakati kuti uve nechokwadi chekuti mhinduro yacho yakanakisa.
Nematanho aya, tinogona kugadzira patani yemhinduro inobva kune derivatives yematambudziko akasiyana-siyana ekushandisa ari pamusoro apa.
Mhedziso
Mienzaniso iri pamusoro inoratidza kuti ma derivatives anogona kushandiswa sei kugadzirisa matambudziko mumamiriro akasiyana-siyana. Kuwana ma high and minimum values, limited optimization, uye mafambiro ekuongorora ingori mashandisirwo mashoma epfungwa yema derivatives. Kuziva matekiniki aya nenzira dzakakosha kune avo vanodzidza masvomhu epamusoro nedzimwe dzidzo dzakabatana.
